Finite Group Databasebeta

← #27278958

Subgroups — #27278958

Order 13568.

500 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284746C211yesmaximal, characteristic
2#20275634C211yesmaximal, characteristic
3C4×#13133141C211yesmaximal, characteristic, complemented
4#20273165C211yesmaximal, characteristic
5#20272616C211yesmaximal, characteristic, complemented
6#20272553C211yesmaximal, characteristic, complemented
7Q8×M16×C53C211yesmaximal, characteristic, complemented
8#16811173C2211yescharacteristic
9#16810941—21no—
10#16810908C411yescharacteristic, complemented
11#16810908C411yescharacteristic, complemented
12#16810823C411yescharacteristic, complemented
13#16810823C411yescharacteristic, complemented
14#16810762—21no—
15C4×#11719924—21no—
16C4×#11719924—21no—
17#16810676C2211yescharacteristic
18C2×#13133143—21no—
19C2×#13133143—21no—
20C2×#13133141—21no—
21C2×C4×#88322C2211yescharacteristic
22C4×C4⋊Dic53C2211yescharacteristic
23#16810385—21no—
24#16810350—22no—
25C2×C4×Q8×C53C2211yescharacteristic
26C53×#446—21no—
27C53×#446—22no—
28C8×Q8×C53—21no—
29C53×#423—21no—
30C53×#423C2211yescharacteristic
31C4×M16×C53—21no—
32C4×M16×C53C2211yescharacteristic
33C2×#11719945—21no—
34C2×#11719924—21no—
35C2×#11719924—21no—
36C22×#88322—21no—
37#13133217C2×C411yescharacteristic
38#13133214C2×C411yescharacteristic
39#13133205—21no—
40#13133191—21no—
41#13133191—21no—
42C4×D212—21no—
43C4×D212—21no—
44C2×C4⋊Dic53—21no—
45C2×C4×Dic53—21no—
46#13133148—21no—
47#13133148—21no—
48C4⋊#88322C2×C411yescharacteristic
49C4⋊#88322C2×C411yescharacteristic
50#13133143—42no—
51#13133143—42no—
52#13133141—24no—
53#13133141—21no—
54#13133141—21no—
55#13133137—21no—
56#13133137—21no—
57C22⋊#88322—21no—
58C4×#88322—21no—
59C4×#88322—21no—
60C4×#88322—21no—
61#13133124—21no—
62C22×Q8×C53D411yescharacteristic
63C2×M16×C53—21no—
64C2×M16×C53D411yescharacteristic
65C2×M16×C53D411yescharacteristic
66C4×Q8×C53—21no—
67C4×Q8×C53—21no—
68C4×Q8×C53C2×C411yescharacteristic
69C4×Q8×C53C2×C411yescharacteristic
70C4×Q8×C53D411yescharacteristic
71C4×Q8×C53D411yescharacteristic
72C2×C4⋊C4×C53—21no—
73C2×C4⋊C4×C53D411yescharacteristic
74C2×C42×C53—21no—
75C2×C42×C53C2311yescharacteristic
76C4⋊C8×C53—22no—
77C4⋊C8×C53—22no—
78C4⋊C8×C53—21no—
79C4⋊C8×C53D412yes—
80#97×C53—22no—
81#97×C53—21no—
82C4×C8×C53—22no—
83C4×C8×C53—21no—
84C2×Q8×C53—21no—
85C2×Q8×C53—22no—
86C2×Q8×C53—41no—
87C2×Q8×C53C22⋊C411yescharacteristic
88C2×Q8×C53C22⋊C411yescharacteristic
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53#4111yescharacteristic
94C2×C22⋊D53—41no—
95C22×Dic53—41no—
96#11719945—41no—
97#11719945—41no—
98#11719945—41no—
99#11719945—41no—
100C2×D212—21no—
101C2×D212—21no—
102C2×D212—21no—
103C2×C4×D53—41no—
104C2×Dic106—21no—
105M16×C53—21no—
106M16×C53—21no—
107M16×C53—21no—
108M16×C53—41no—
109M16×C53—41no—
110M16×C53SD1612yescomplemented
111C2×C8×C53—21no—
112C2×C8×C53—21no—
113C2×C8×C53—22no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53—21no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53C22⋊C411yescharacteristic
120C4⋊C4×C53C22⋊C411yescharacteristic
121C42×C53—21no—
122C42×C53—21no—
123C42×C53—21no—
124C42×C53—21no—
125C42×C53C22×C411yescharacteristic
126C42×C53C2×D411yescharacteristic
127C42×C53C2×D411yescharacteristic
128C42×C53#4111yescharacteristic
129#11719924—42no—
130#11719924—42no—
131#11719921—41no—
132#11719921—41no—
133C4⋊Dic53—22no—
134C4⋊Dic53—21no—
135C4⋊Dic53—21no—
136C4⋊Dic53—41no—
137C4×Dic53—41no—
138C4×Dic53—41no—
139C2×#88322—22no—
140C2×#88322—21no—
141C2×#88322—21no—
142C2×#88322—41no—
143C2×#88322—41no—
144C23×C53#12111yescharacteristic
145C22×D53—41no—
146Q8×C53—21no—
147Q8×C53—21no—
148Q8×C53—21no—
149Q8×C53—42no—
150Q8×C53#10412yescomplemented
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—41no—
158C2×C4×C53—42no—
159C2×C4×C53—81no—
160C2×C4×C53C2×C22⋊C411yescharacteristic
161C2×C4×C53C4×D411yescharacteristic
162C2×C4×C53C4×D411yescharacteristic
163C2×C4×C53#12011yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12311yescharacteristic
166C2×C4×C53C2×SD1611yescharacteristic
167C2×C4×C53#13511yescharacteristic
168C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C53#13711yescharacteristic
170C22⋊D53—81no—
171C22⋊D53—81no—
172C2×Dic53—42no—
173C2×Dic53—41no—
174C2×Dic53—81no—
175C2×Dic53—81no—
176D212—42no—
177D212—41no—
178D212—41no—
179D212—41no—
180D212—41no—
181C4×D53—81no—
182C4×D53—81no—
183Dic106—41no—
184Dic106—41no—
185C8×C53—22no—
186C8×C53—41no—
187C8×C53—42no—
188#88322—42no—
189#88322—41no—
190#88322—41no—
191#8992—531nomaximal, Sylow
192C22×C53—21no—
193C22×C53—21no—
194C22×C53#38611yescharacteristic
195C22×C53#46011yescharacteristic
196C22×C53#46411yescharacteristic
197C2×D53—81no—
198C2×D53—81no—
199C2×D53—81no—
200C4×C53—21no—
201C4×C53—22no—
202C4×C53—21no—
203C4×C53—21no—
204C4×C53—41no—
205C4×C53—41no—
206C4×C53—41no—
207C4×C53C2×#10411yescharacteristic
208C4×C53#42111yescharacteristic
209C4×C53C4×SD1611yescharacteristic
210C4×C53#44011yescharacteristic
211C4×C53#44811yescharacteristic
212C4×C53#44911yescharacteristic
213Dic53—82no—
214Dic53—81no—
215#3103—531no—
216Q8×M16D5311yescharacteristic, complemented
217#1980—531no—
218C4×#103—531no—
219#1492—531no—
220#1303—531no—
221#1278—531no—
222C2×C53—21no—
223C2×C53—21no—
224C2×C53#187111yescharacteristic
225C2×C53#188011yescharacteristic
226C2×C53#188111yescharacteristic
227D53—161no—
228#532—531no—
229C2×C4×Q8C2×D5311yescharacteristic
230#481—1061no—
231#475—531no—
232#475—531no—
233#460—531no—
234#460—531no—
235#448—1061no—
236#446—21no—
237#446—22no—
238C8×Q8—21no—
239C4×SD16—1061no—
240C4×SD16—1061no—
241#431—531no—
242#423—21no—
243#423C2×D5311yescharacteristic
244C2×#104—1061no—
245C2×#104—1061no—
246C2×#103—1061no—
247C4×M16—21no—
248C4×M16C2×D5311yescharacteristic
249C2×C4×C8—531no—
250C4×C4⋊C4—531no—
251#340—1061no—
252D4⋊C8—1062no—
253C53#899211yescharacteristic, complemented, Sylow
254C2×#41—1061no—
255C22×Q8D21211yescharacteristic
256C2×SD16—1061no—
257C2×SD16—1061no—
258C2×M16—21no—
259C2×M16C22⋊D5311yescharacteristic
260C2×M16C22⋊D5311yescharacteristic
261C22×C8—1061no—
262C4⋊Q8—531no—
263#127—531no—
264#124—1061no—
265#121—1061no—
266#121—1061no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8C4×D5311yescharacteristic
270C4×Q8C4×D5311yescharacteristic
271C4×Q8C22⋊D5311yescharacteristic
272C4×Q8C22⋊D5311yescharacteristic
273C4×D4—1061no—
274C4×D4—1061no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—1061no—
277C2×C4⋊C4D21211yescharacteristic
278C2×C42—21no—
279C2×C42—1061no—
280C2×C42C22×D5311yescharacteristic
281#106—1061no—
282#106—1061no—
283C4⋊C8—22no—
284C4⋊C8—22no—
285C4⋊C8—21no—
286C4⋊C8—531no—
287C4⋊C8—531no—
288C4⋊C8C22⋊D5312yes—
289#104—2122no—
290#104—2122no—
291#103—1064no—
292#103—1061no—
293#103—1061no—
294#102—1061no—
295#102—1061no—
296C22⋊C8—1061no—
297#97—22no—
298#97—21no—
299C4×C8—22no—
300C4×C8—21no—
301C4×C8—1061no—
302C4×C8—1061no—
303C4×C8—1061no—
304#95—1061no—
305#41—2121no—
306#41—2121no—
307#41—2121no—
308#41—2121no—
309C2×Q8—21no—
310C2×Q8—22no—
311C2×Q8—41no—
312C2×Q8—1061no—
313C2×Q8#1171992111yescharacteristic
314C2×Q8#1171992111yescharacteristic
315C2×D4—1061no—
316C2×D4—1061no—
317C2×D4—1061no—
318C2×D4—2121no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—2121no—
322C22×C4—2121no—
323C22×C4C2×D21211yescharacteristic
324C22×C4#1171994511yescharacteristic
325C22×C4C2×C22⋊D5311yescharacteristic
326SD16—2122no—
327SD16—2122no—
328M16—21no—
329M16—21no—
330M16—21no—
331M16—41no—
332M16—41no—
333M16#1171992412yescomplemented
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—22no—
337C2×C8—1062no—
338C2×C8—1061no—
339C2×C8—1061no—
340C2×C8—2121no—
341C2×C8—2121no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—1062no—
348C4⋊C4—1061no—
349C4⋊C4—1061no—
350C4⋊C4—2121no—
351C4⋊C4#1171992111yescharacteristic
352C4⋊C4#1171992111yescharacteristic
353C22⋊C4—2121no—
354C22⋊C4—2121no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—2121no—
360C42—2121no—
361C42C2×C4×D5311yescharacteristic
362C42#1171994511yescharacteristic
363C42C2×C22⋊D5311yescharacteristic
364C42C2×C22⋊D5311yescharacteristic
365C23—2121no—
366C23#1313319111yescharacteristic
367Q8—21no—
368Q8—21no—
369Q8—21no—
370Q8—42no—
371Q8—2121no—
372Q8—2121no—
373Q8#1313314312yescomplemented
374D4—4241no—
375D4—4241no—
376D4—2122no—
377D4—2121no—
378D4—2121no—
379D4—2121no—
380D4—2121no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—81no—
390C2×C4—4241no—
391C2×C4—4241no—
392C2×C4—4241no—
393C2×C4—4241no—
394C2×C4—2122no—
395C2×C4—2121no—
396C2×C4C2×#1171992111yescharacteristic
397C2×C4C4×C22⋊D5311yescharacteristic
398C2×C4C4×C22⋊D5311yescharacteristic
399C2×C4#1313318611yescharacteristic
400C2×C4#1313319111yescharacteristic
401C2×C4#1313320011yescharacteristic
402C2×C4C2×#1171992411yescharacteristic
403C2×C4#1313323711yescharacteristic
404C2×C4#1313324211yescharacteristic
405C2×C4#1313324811yescharacteristic
406C8—22no—
407C8—41no—
408C8—42no—
409C8—2122no—
410C8—2121no—
411C8—2121no—
412C22—21no—
413C22—21no—
414C22—4241no—
415C22—4241no—
416C22—4241no—
417C22#1681050211yescharacteristic
418C22#1681082311yescharacteristic
419C22#1681084511yescharacteristic
420C4—21no—
421C4—22no—
422C4—21no—
423C4—21no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—4242no—
428C4—4241no—
429C4C2×#1313314311yescharacteristic
430C4#1681063911yescharacteristic
431C4C4×#1171992411yescharacteristic
432C4#1681072111yescharacteristic
433C4#1681076211yescharacteristic
434C4#1681076811yescharacteristic
435C2—21no—
436C2—21no—
437C2—8481no—
438C2#2027501711yescharacteristic
439C2#2027507011yescharacteristic
440C2#2027507711yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingQ8×M16×C53C2
socleC22×C53#386
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenterQ8×M16D53
hypocenterC53#8992
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ Q8×M16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C4×#13133141C2semidirect
5. #20272616C2semidirect
6. #20272553C2semidirect
7. Q8×M16×C53C2semidirect
10. #16810908C4semidirect
11. #16810908C4semidirect
12. #16810823C4semidirect
13. #16810823C4semidirect
110. M16×C53SD16semidirect
150. Q8×C53#104semidirect
216. Q8×M16D53semidirect
253. C53#8992semidirect
333. M16#11719924semidirect
373. Q8#13133143semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202847462
#202756342
C4×#131331412
#202731652
#202726162
#202725532
Q8×M16×C532
#899253